The Sharkovsky ordering describes the coexistence of cycles with different periods for
discrete-time dynamical systems given by maps \( f: I \to I\ ,\) where \(I\) is an interval in the real line \(\mathbb{R}\ ,\) and, possibly \(I=\mathbb{R}\ .\)
One can also say that it provides a forcing relation for the existence of cycles
of certain periods due to the presence of a cycle of another period.

Theorem * (Sharkovsky 1964) If a continuous map of an interval into itself has a cycle of period \(m\ ,\) then it has a cycle of any period \({\tilde m} ~\prec~ m\ .\) Moreover, for any \(m\) there exists a continuous map that has a cycle of period \(m\) but does not have cycles of periods \({\overline m}\ ,\) \(m ~\prec~ {\overline m}\ .\)

This theorem also shows how cycles of different periods can be arranged on \(I\ .\)
If \(B\) is a cycle, let \(S(B)\) be the interval
\([\min \{x \in B\}, \max \{x \in B\}]\ ,\) referred to as the support of the cycle.
If \(m\) is the period of the cycle \(B\) and \({\tilde m}\) is any number such that \({\tilde m} ~\prec~ m\ ,\) then
the map \(f\) also has a cycle \({\tilde B}\) of period \({\tilde m}\) such that \(S({\tilde B}) \subset S(B)\ .\)
Indeed, instead of the map \(f\ ,\) one can consider a continuous map \(f_B\) that coincides with \(f\) on \(S(B)\) and equals const outside the interval \(S(B)\ .\) The theorem remains true for \(f_B\ ,\) in particular, \(f_B\)
has cycles of period \({\tilde m}\ ,\) but all cycles of \(f_B\) are in \(S(B)\ .\)

The ordering (1) can be interpreted in terms of stratification (Block, Coppel 1992).
Let \(C(I,I)\) denote the set of all continuous maps of \(I\) into itself
and \({\mathbb P}_n\) be the subset of \(C(I,I)\) consisting of maps
which have cycles of period \(n\ .\)
According to (1), if \(m ~\prec~ {\overline m}\)
then \({\mathbb P}_m \supset {\mathbb P}_{\overline m}\ .\)
Hence, \({\mathbb P}_1 \supset {\mathbb P}_2 \supset {\mathbb P}_4 \supset ...
\supset {\mathbb P}_5 \supset {\mathbb P}_3\ .\)

The ordering (1) has a property of stability (Block 1981):
if \(f\) has a cycle of period \(m\ ,\) then there exists \(\varepsilon > 0\)
such that whatever \({\tilde m} ~\prec~ m\ ,\)
any map \({\tilde f}\,: \sup_{x \in I} |{\tilde f}(x) - f(x)| < \varepsilon\)
has a cycle of period \({\tilde m}\ .\)

The following important corollary of the theorem relates to bifurcation theory:
if the map \(f\) depends on a parameter, the ordering (1) also
gives a universal ordering for the birth of cycles of new periods
when this parameter varies. For example, the bifurcation diagram for the logistic family
of maps
\[\tag{2}
x \mapsto \lambda x(1-x) \;,
\]

shown in Figure 1, displays
the birth of attracting cycles whose periods are ordered according to (1), when
\(\lambda\) increases from 2.9 up to 4.
At first, there is an attracting cycle of period 1 (fixed point),
at \(\lambda = 3\) an attracting cycle of period 2 is created, later
period \(2^2\ ,\) period \(2^3\) cycles, etc. appear;
a period 3 cycle first appears when \(\lambda = 1 + 2 \sqrt 2 \approx 3.83\ .\)
If \(\lambda_{n}\) denotes the parameter value
corresponding to the birth of the first cycle of period \(2^{n}\ ,\)
then, as noticed by Feigenbaum, Coullet, and Tresser,
\[
\delta_{n} = (\lambda_{n}-\lambda_{n-1})/(\lambda_{n+1}-\lambda_{n})
\rightarrow \delta = 4.66920... \qquad {\rm as}~~~ n \rightarrow \infty,
\]
that is, the rate of appearance of cycles of double periods is characterized
by the number \(\delta\) which is often called the Feigenbaum constant.
It turns out that not only the sequence of bifurcations, defined by (1),
but also the rate of bifurcations, defined by the constant \(\delta\ ,\)
are "universal" in the sense that they are valid for the whole class
of differentiable maps (and not only the logistic family).

There are maps having only cycles of periods \(2^n, ~n=0,1,2,...\ ;\)
an example is the logistic map (2) with
\(\lambda = \lambda^* = \lim_{n \to \infty} \lambda_{n} \approx 3.57\ ,\) which is
the limit of maps having cycles of periods \(2^n,~n < \infty,\) only ( Figure 1).
Maps of this kind are called maps of the type \(2^\infty\ ;\) the first
example of such a (non-smooth) map was probably given by Sharkovsky (1965).
Sometimes the ordering (1) of natural numbers is supplemented
with the symbol \(2^\infty\ ,\) inserted after all numbers of the form \(2^n\)
and before the numbers \(\not= 2^n\ .\)

Proofs of Sharkovsky theorem

The proof of the theorem is based on the intermediate value
theorem and actually uses only the fact that if \(f\) is a
continuous map and \(J\) is an interval such that \(f(J) \supset J\)
then on \(J\) there exists a fixed point of the map \(f\ .\)

Since the end of 1970's, there have been published many papers
with various proofs of the theorem or its parts, as well as proofs
of the theorem for special classes of maps. All these can be
divided into three directions:

Improvement of the original proof (see, for example, Du 2004, Du 2007).

The map in Fig.2 has a cycle of period 3, that consists of the
points \(b_1, b_2, b_3\ .\) For this cycle, \(\pi = \left(
\begin{array}{ccc} 1&2&3 \\
2&3&1 \end{array}\right)\) and \(f(\Im_1) \supset \Im_2,
f(\Im_2)\supset\Im_1 \cup \Im_2\ ,\)
so that the transition matrix has the form \(\left(
\begin{array}{cc} 0&1 \\ 1&1 \end{array}\right)\ ,\)
and the graph of admissible transitions is as in Fig.2

Figure 2: Map having a cycle of period 3 (left) and the graph of admissible transitions responding to this cycle (right).

By analyzing the matrix or the graph of transitions, one can show
the existence of cycles of various periods of the map. For
example, if we use the alphabet consisting of the symbols
\(a_1,a_2, ...,a_{m-1}\ ,\) then to each symbolic sequence \(a_{r_1},
a_{r_2}, ...,a_{r_j},a_{r_{j+1}}, ..., 1\leq r_j\leq m-1\ ,\)
admissible by the matrix of transitions (i.e., \(\mu _{r_jr_{j+1}}
= 1\) for \(j = 1, 2, ...\)), there corresponds at least one orbit
going through the intervals \(\Im_1, \Im_2, ..., \Im_{m-1}\)
in the order \(\Im_{r_1} \to \Im_{r_2} \to ... \to \Im_{r_j} \to
\Im_{r_{j+1}} \to....\) (for any \(j > 1\ ,\) the set \(\Im_{r_1 r_2 ...
r_j} = f^{-1} (f^{-1} ... (f^{-1} \Im_{r_j} \cap \Im_{r_{j-1}})
\cap ... \cap \Im_{r_{2}}) \cap \Im_{r_{1}}\) is not empty and
hence each point \(x \in \Im_{r_1 r_2 ... r_j}\) gives rise to
an orbit that visits sequentially the intervals \(\Im_{r_1},
\Im_{r_2}, ..., \Im_{r_j}\)). In addition, if the symbolic sequence
is periodic with the least period \(n\ ,\) then the system has at
least one cycle of period \(n\) (going through the intervals
\(\Im_1, \Im_2, ..., \Im_{m-1}\) in this particular order).

For the map having a cycle of period 3 (Fig.2), the periodic
sequence of transitions \(\Im_{1} \to \Im_{2} \to \Im_{2} \to ...
\to \Im_{2} \to \Im_{1}\) of any length is admissible, hence such
a map has a cycle of any period. However, the proof of the theorem in general case requests more significant efforts and it makes sense to refer, for example, to
(Block, Coppel 1992, Alseda et al. 2000).
The first proofs of such kind were given by
Straffin (1978), Block et al. (1980), Ho and Morris (1981), and Burkart (1982).

To prove the second part of the theorem, we need only to find a map
having a cycle of period \(m\) and not having cycles of periods
\({\tilde m}\ ,\) \(m ~\prec~ {\tilde m}\ .\) To do this, we can use the
map \(g\,: x \mapsto 4x(1-x)\ ,\) which is known to have cycles of all
periods. For any value of \(m\ ,\) the map has a finite number of
cycles of period \(m\) (not exceeding \(2^m\,/\,m\)). Out of all the
cycles of period \(m\) let us choose the cycle \(B_m\) whose support
\(S(B_m)\ ,\) i.e., the interval \([\min \{x \in B_m \}, \max \{x \in B_m \}]\ ,\)
is of the smallest length (if there are a few cycles like this
one, take any of them). Then the continuous map
\[
g_{(m)}\,: x \mapsto \left\{\begin{array}{ccc}
g(x), & x \in S(B_m), \\
const, & x \notin S(B_m),
\end{array}\right.
\]
has a cycle of period \(m\) (namely, the cycle \(B_m\)) and does not
have cycles of period \({\tilde m}\ ,\) \(m ~\prec~ {\tilde m}\ .\)

History

The fact that \(1 ~\prec~ 2\) is merely a consequence of the
intermediate value theorem in classical analysis.
Even Henri Poincare used this fact to prove that a second-order
ordinary differential equation has an equilibrium point
in a region of the phase plane, that is bounded by a closed curve –
periodic orbit of the equation.

The next step in the description of the coexistence of cycles was
the proof that \(2 ~\prec~ m\) when \(m>2\ .\)
From works (Leibenzon 1953, Coppel 1955, Myshkis, Lepin 1957),
devoted to conditions of convergence for function sequences of the
form \(f^n(x), n=0,1,2,..., x\in I\ ,\) it was at once evident
that if a map does not have cycles of period 2,
then it does not have cycles of greater periods. In (Sharkovsky 1961)
it was proven that if a map has a cycle of period \(\neq
2^{l}, l=0,1,2,...\ ,\) then it has cycles of periods \(2^{i},
i=0,1,2,...\ .\)

Sharkovsky Theorem and its complete proof was published in 1964 in
Ukranian Mathematical Journal (Sharkovsky 1964)
in Russian (the article itself was submitted to the journal in
March of 1962); the second assertion of the theorem was given in
the article in the form of examples. An English translation of the
paper was published in 1995 in the International Journal of
Bifurcation and Chaos (Sharkovsky 1964).
However, as early as 1977, the complete presentation of Sharkovsky’s (1964)
results (including the fact that between any two points of a cycle there
exists at least one point of a cycle of smaller period) was
published in English in (Stefan 1977).

When the Sharkovsky Theorem was first published, it was reasonable to
"put 3 at the head". Now, in the mathematical literature, one can
find both versions, the "forward" ordering, i.e., in the
form (1), and the "reverse" ordering, which begins with 3 and ends
with 1. The forward ordering (1) is likely to be
preferred, since it more closely resembles the standard ordering
of natural numbers and since it characterizes the evolution of
systems in the direction from simple to complex (at least
with respect to the set of existing cycles), and not the reverse:
from complex to simple.

The author was trying to popularize these results as far back as in 1960s.
He made presentations on V.V.Nemitsky's seminar in
Moscow State University in 1963-1964; at about the same time he
described these results to L.V.Keldysh in the Steklov Mathematical
Institute in Moscow.

On the 4th International Conference on Nonlinear Oscillations
(Prague 1967), the author reported his results concerning the
coexistence of periodic solutions for the difference equation
\(x(n+1)=f(x(n))\ .\) However, the organizers of the conference
included only the abstract of the report into the proceedings of
the conference, published in 1968 (Sharkovsky 1968).
The abstract could be considered as the first publication of the
ordering (1) in English (!), not counting the shorter
English-language abstract in (Sharkovsky 1964).
Ya. Kurzweil, one of the greatest Czech mathematicians,
later tried to justify the decision of the organizers pointing
out that they, being for the most part mechanicians, did not
understand how much this "simple" difference equation would be
pertinent to the theory of oscillations.

Widespread interest (not only among mathematicians but among
experts in various fields of natural science as well) in
one-dimensional dynamical systems, in particular, in the
coexistence of cycles
have been spurred by the article of Li and Yorke (1975).

The history of studies relevant to the theorem on the coexistence
of cycles is described in sufficient detail in (Alseda et al. 2000,
Misiurewicz 1997).

Generalizations

The ordering (1) has been attracting attention of many mathematicians.
Not only have they suggested new
versions of the proof, but also they investigated
the possibility of extending the results to more complex structures (beyond
cycles), to wider classes of maps (discontinuous,
multi-valued, random, and so on), to different types of phase spaces (one-dimensional: circle, stars, graphs, so-called
hereditary decomposable chainable continua, or even
multi-dimensional and infinite-dimensional, but for a special
class of maps, such as triangular, cyclic, and so on). These
studies led to the appearance of a new section in the
Mathematics Subject Classification of AMS, namely, the section
"37E15 – Combinatorial Dynamics", in 2000.

Thus of the questions regarding possible
extensions and generalizations of Sharkovsky Theorem, the first
is: For which classes of maps and topological spaces do theorems
similar to the one above hold?

One example of this question is a generalization to non-continuous interval maps.
Recently Szuca (2003) proved that the ordering (1) holds
even for (non-continuous) maps whose graphs are connected
\(G_\delta\) sets (of the plane). Andres with colleagues (2002)
suggested a multivalued version of the theorem. Kluenger (2001), Andres (2008)
considered possibilities to extend the theorem on random maps.

There have also been fairly complete studies of circle maps,
i.e., continuous maps of a circle to itself. The rigid rotation
\(x\rightarrow x+\alpha \mod 1\ ,\) a classic circle map, lacks cycles
of any periods when the rotation angle \(\alpha\) is irrational; for
rational \(\alpha=p/q\ ,\) it has cycles of period \(q\) only. However,
if a circle map is assumed to have cycles of different periods,
then the problem of coexistence of cycles becomes worthy of
attention (Efremova, Rakhmankulov 1980, Block et al. 1980, Alseda et al. 2000).

The interval and circle are one-dimensional manifolds, which could
be treated as simplest connected graphs. There are theorems on
coexistence of cycles of different periods for continuous maps of
graphs of various types. There the ordering of periods is distinct
from (1); moreover, it depends on the type of graph, see, for
example, (Alseda et al. 2000, Misiurewicz 2001).
This brings up the question: For which topological spaces does the
ordering (1) hold? Among these spaces are, for instance,
so-called "hereditary decomposable chainable continua"
(Schirmer 1985, Ingram 1988, Minc, Transue 1989, Alcaraz, Sanchis 2003).

The ordering (1) is a "one-dimensional phenomenon". For any subset
of the natural numbers that contains 1, there exists a
continuous (analytical) map of the plane into itself, which has
this subset as the set of periods of its cycles (Lisovyk 1985).

On the other hand, even in \(R^n\) there is a class of maps for
which the ordering (1) holds; these are so-called "triangular"
maps (or skewed products of one-dimensional maps) (Kloeden 1979):
\[ x_1 \mapsto f_1(x_1)~~~~~~~~~~~~~ \]
\[ x_2 \mapsto f_2(x_1, x_2)~~~~~~~~~ \]
\[ \ldots \]
\[ x_n \mapsto f_n(x_1, x_2, ... , x_n). \]
Another example of special classes of continuous maps is so-called the cyclic map
\[ x_1 ~~\mapsto ~~f_1(x_2)~ \]
\[ \ldots \]
\[ x_{n-1} ~\mapsto ~~f_{n-1}(x_n) \]
\[ x_n ~~\mapsto ~~f_n(x_1).~ \]
For such a map the coexistence of periods can be described too
(Balibrea, Linero 2001)
but in a form different from (1).

Of course, these multidimensional maps are close to one-dimensional
ones in one sense or another. In (Zgliczynski 1999) is considered
"direct" multidimensional but small perturbations of one-dimensional maps.

Finally, there are even infinite-dimensional dynamical systems
in which the (co)existence of periodic orbits
is controlled by the ordering (1).
First of all, among these are dynamical systems generated by (scalar)
difference equations \(x(t+1) = f(x(t))\) with continuous time
(\(t \in {\mathbb R}^+\)) and
also by boundary value problems reducible to such difference equations
(Sharkovsky, Sivak 1994, Sharkovsky 1995, Sharkovsky et al. 2006).

The second question relevant to the generalization and extension
of Sharkovsky's Theorem is the question of the coexistence of
structures other than cycles, i.e., invariant sets consisting of
more than one orbit (cycle) or even of infinity of orbits. An
example of such a structure is a minimal set different from a cycle;
in the case of the real line, these minimal sets are Cantor sets.

One can introduce the notion of period for minimal sets.
The period is the least natural \(m\) (if it exists) such that
a minimal set is portioned into \(m\) non-overlapping sets minimal
with respect to the map \(f^m\ ,\) which therefore make a cycle
(of sets) of period \(m\) with respect to the map \(f\ .\)
The least \(m\) does not exist when
the minimal set turns to be infinitely divisible (as the one
appearing after the infinite sequence of period-doubling
bifurcations). The order of coexistence for Cantor minimal sets of
special types is described in (Ye 1992).

Notice that the order of coexistence for homoclinic trajectories
is also similar to the ordering (1)
(Fedorenko, Sharkovsky 1982).

An additional question relates to the means for describing the
coexistence of cycles or sets. So far we have used the notion of
period, which is characterized by a natural number. But maps can
have many cycles of some fixed period \(m\ .\) For example, the map
\(x\rightarrow 4x(1-x)\) has \((2^{m}-2)/m\) cycles of period \(m\) with
\(m\) being any prime number. Hence, it makes sense to use sometimes
a characteristic of a cycle more precise than just its period. In
the case of the real line, we can pay attention to the relative
position of cycle points on the line and hence use the cyclic
permutations \(\pi\) mentioned above. As a result, there arises a
problem of coexistence (or forcing) of cycles of different types,
i.e., cyclic permutations corresponding to these cycles.

Let "\(\hookleftarrow\)" be the order relation defined as\[\pi \hookleftarrow \pi'\] if and only if every continuous map
of an interval, that has a cycle of type \(\pi'\ ,\) also has a
cycle of the type \(\pi\ .\)
The set of all cyclic permutations with this order relation
is not a linearly ordered set because the order "\(\hookleftarrow\)"
is only a partial order (Fedorenko 1986, Baldwin 1987).
For example, there are only three cyclic permutations of length 4
(and three inverse to them)
\[\pi_1 = \left(
\begin{array}{cccc}
1&2&3&4\\
3&4&2&1 \end{array}\right)\ ,\] \(\pi_2 = \left(
\begin{array}{cccc}
1&2&3&4\\
2&3&4&1 \end{array}\right)\ ,\) and \(\pi_3 = \left(
\begin{array}{cccc}
1&2&3&4\\
3&1&4&2 \end{array}\right)\ ,\)
and it is easy to check that \(\pi_1 \hookleftarrow \pi_2\) and \(\pi_1 \hookleftarrow \pi_3\ ,\)
but \(\pi_2, \pi_3\) are noncomparable.
Besides, the permutation \(\pi_3\) cannot be realized by unimodal maps.
However, the order relation \(\hookleftarrow\) is linear on the set
of cyclic permutations generated by unimodal maps. This result can be
obtained, e.g., as a corollary of the linear ordering for the codes
of trajectories, which used in kneading theory (Sharkovsky et al. 1997).

Notice that among various cyclic permutations of length \(m\) one can
select so-called minimal permutations corresponding to the cycles
of period \(m\) that are the first to appear via bifurcations (or
equivalently, to the cycles of period \(m\) that do not cause the
existence of cycles of other types but of the same period). These
cycles are sometimes called minimal.

Minimal permutations of length \(m\) have the following form
(Sharkovsky 1964, Stefan 1977) :

When \(m=2k\ ,\) these are permutations \(\pi\) for which the following inductive property holds: The sets \(N_1 = \{1,2,...,k\}\) and \(N_2 = \{k+1,k+2,...,2k\}\) are invariant with respect to \(\pi^{2}\) (i.e., if \(i \in N_s\) then also \(\pi^2 (i) \in N_s, ~s=1,2\)), and the restriction \(\pi^{2}\) on \(N_1\) is the minimal permutation.

If \(\pi\) is a minimal permutation and \(b_1 < b_2 < ... < b_m\) are
some points on the real line, then the continuous piece-wise
linear map \(f_{\pi} :[b_1,b _{m}]\rightarrow[b _{1},b _{m}]\) such
that \(f_{\pi}(b_i) = b _{\pi(i)}, ~i = 1, ..., m\ ,\) and \(f_{\pi}\)
is linear on \([b _{i},b _{i+1}], ~i = 1, ..., m-1\ ,\) has a cycle of
period \(m\) (formed by the points \(b_1,b_2, ..., b_m\)) and does not
have cycles of periods \({\tilde m}\ ,\) \(m ~\prec~ {\tilde m}\ .\)

Besides the classification of cycles with respect to periods and
types, there is another classification that uses so-called
rotation numbers for cycles – certain rational numbers. A variant
of such a classification is the following:
if \(b_1 < b_2 < ... < b_{m}\) are points of a cycle \(B\) of period \(m\)
and exactly \(p\) of them are such that \(f(b)<b\ ,\)
then the number \(p/m\) is referred to as the rotation number of the cycle \(B\ .\)
Articles (Blokh, Misiurewicz 1997, Bobok, Kuchta 1998)
describe the coexistence of cycles of interval map with different
rotation numbers.